The first time I encountered this text, I remember being quite disappointed. And I LOVE the three books that come before it. But in trying to figure out what didn’t quite sit right with me back then, I find myself wondering that now. Firstly, compared to his other books, real page turners if you ask me, this one I found boring, lifeless, lackluster. I remember the first time I encountered Intensive Science and Virtual Philosophy, it was electrifying. But this book seemed to just restate the obvious when it came to his social analyses, but then there were the opening sections on terms/entities and relations. This was before Object Oriented Philosophy had come on the scene. But I remember something about this whole approach didn’t sit right with me then, and still doesn’t sit right with me now, despite the fact that this all comes from Deleuze, one of my philosophical heroes. What gives, then or now?

Relations versus Terms: Deleuze’s Hidden Polemics

Firstly, I think its important to keep in mind that Deleuze’s insistence that relations are external to their terms is a classic example of what Bakhtin calls ‘hidden polemic.’ He’s arguing against Lacan as the proximate enemy, Hegel as the shadowy figure behind him, and a variety of other ghost-like figures lurking in the wings, including deconstructionists, certain redeployments of medieval philosophy in France at the time, etc. As I’ve argued elsewhere, I think Deleuze’s famous hatred of all things Hegel is because of his own anxiety of influence – to me, Deleuze often out-Hegel’s Hegel why saying that the one thing he is not is Hegelian. A controversial argument to make, but for more see my post on this in regard to Hegel’s Logic and Deleuze’s ‘Logic’, or the Cinema books.

So I think we need to take Deleuze’s ‘relations are exterior to their terms’ in regard to its historico-discursive context. To hypostatize it from this is to miss out on half of what he was trying to say. As Bakhtin argues, hidden polemic involves an unmentioned opponent which haunts the original text, a ‘he/she that must not be named,’ so to speak. Unless one figures out who the opponent is, you are literally missing half of the context for reading the original text. The result is you render the original extimate from itself (or more extimate than it was originally). Reading Foucault, for example, is a constant attempt to read his hidden polemic, because he rarely does more than hint at who he is arguing against when going on an extended excursion on some seemingly obscure and arcane matter in the midst of an otherwise seemingly straightforward argument.

But when it comes down to it, as much as I like MOST of what Deleuze has to say, and I see WHY he said what he did about terms and relations at the time, I’m not sure I buy it. Nor do I think a good Deleuzian should. For a thinker so against any firm binary, the binary he establishes here between terms and relation is far too neat for Deleuze. The only reason why I think he keeps it somewhat consistently is because he is being polemical, which means, in this case, he is not being just a good Deleuzian. He is sparring in a manner which distorts the text, the text does something other than what it seems to be doing. Of course, all texts do this. But this binary, one of the few that Deleuze does not dissolve shortly after he produces it, seems out of character to me.

Fuzzy Set Theory, Or, How a Barber Can Sorta Shave Himself . . .

Part of this is mathematical. Deleuze was quite aware of set theory. But he rarely mentions what today we would call ‘fuzzy’ set theory. Deleuze in fact tends to stay away from set theory, largely I think because it is generally a black-and-white type affair, which is precisely why Lacan and later Badiou have found it so useful. But fuzzy set theory, now that’s a different story.

One area where fuzzy set theory differs from traditional set theory is that it is possible for sets to partially include themselves. In classical set theory, an elemen either is or is not included in a set – a pea is green, a cherry is not. Something either is or is not green, its that simple. But as anyone who lives in the real world knows, the world is fuzzy, and so sets must be also – it is completely possible for a pea to be partially green. In classical set theory, one gets either/or, 1 or 0. But in fuzzy set theory, you have all the values in between. It is in this manner that fuzzy set theory is much closer to natural language, and is much better at modeling these sorts of mixed states of affairs.

Classical set theory famously foundered on the paradox articulated by Bertrand Russell, the so-called ‘Russell’s paradox’, often described by the riddle, ‘If everyone in town is shaved by the barber, does the barber shave himself?’ Any attempt to parse this paradox leads to either incompleteness or inconsistency – if somebody else shaves the barber, then they are the barber, unless we have changed the meaning of the word, but then, is the barber who is shaved still the barber when being shaved? Or can the barber shave himself, in which case, does he split himself into half barber and half barbee? The very category of barber stops making sense when we reach this point of recursion, or, we need to bring in another person, a new barber, but this just shifts the burden to the new barber.

Frege’s dream of founding math on its own axioms, of deducing the axioms of math from themselves, foundered on Russell’s famous objection, though it was left to Goedel to put this all into formal mathematical language as a proof. But to return to set theory, we see here precisely the limits of classical set theory – a set cannot include itself, the set itself is extimate to itself. This extimacy of the set to itself is the foundation of the Lacanian notion of the S1, the constitutive exclusion, that which links Lacan to Schelling (the most radical thinker of the extimacy of the ground as exists in the Western philosophical tradition) as much as to Hegel (‘the spirit is a bone’).

But fuzzy set theory presents a way out of this alternative. In classical set theory, membership of an element in a set is often counted, we can assign the value of 1 to the statement ‘Set A includes Set B’ if this is true, and 0 if false. Nice and binary. But fuzzy math is based on the notion that everything can exist between states of true and false, 1 and 0. So, in fuzzy set theory, there are DEGREES of inclusion. Set A may include Set B to a value of .2. What could this mean?!

If two sets completely include each other equally, to a value of 1/2 or .5, we have a case in which they both include each other. For example, if the Republican and Democratic party decide to merge, we can say that all Republicans are now Democrats, all Democrats now Republicans. Its not senseless to say they are both the same and different to equal degrees, a situation of both-and, which is distinctly different from indistinction. We are not saying there is one party, but rather, a composite unity. The only way to properly represent this is by fuzzy set theory. But let us say that this new party still leans to the Republican side, favors Republican values, despite being equal on paper. Then perhaps we could say that the Republican Set includes the Democratic one to a degree of .8, while the Democratic Set includes the Republican set to a degree of .2. This is how fuzzy sets allow for partial inclusion.

If we move from static to dynamic situations, we end up within the realm of topological shapes. Classical set theory finds its physical analogue, not in geometry, but in what is known as ‘point-set topology’, a branch of abstract math which imagines sets as semi-physical objects, composed of sets laid out in abstract space. Venn Diagrams are extreme simplifications of how point-set topology works – point-set topology is in some ways a diagrammatics for logic.

But fuzzy set theory is much more difficult to diagram. For in fact, we end up in the realm of non-orientable figures, such as the Moebius strip or Klein bottle. Only these figures can be said to include themselves. The Moebius strip, for example, is a two dimensional figure that includes itself to a degree of .5 or 1/2, for in fact, it both is and is not a one-sided surface. The ‘twist’ that one installs when one wants to create a real life Moebius strip is precisely what is at stake with the 1/2 value of self-inclusion when creating a fuzzy set. A Klein bottle is simply a higher dimensional analogue of the Moebius strip. But might it be possible to imagine a Moebius strip whose self-inclusion value is not an even .5, but say, a .2 or .3?

Mathematically, there is no reason why not. Making a physical shape of this might be difficult (how would you install a partial twist?!), but just because you can’t imagine in 3D how to construct something is not something which has ever bothered mathematicians. But it seems to me that we do this with natural language all the time. For example, to what extent do terms such as ‘animals’ and ‘critters’ include each other? It seems to me that fuzzy set theory provides the only way to logically describe certain very everyday aspects of natural language.

Relations Versus Terms? Yes Please!

Why this long digression when discussing the binary between terms and relations? The fact is that I feel this binary doesn’t sit well with me. For in fact, as Object Oriented Philosophy has argued, ALL objects exceed their relations. But I’d also like to argue that ALL RELATIONS ALSO exceed their terms. For is not each relation something which is so unique that it must withdraw also? Aren’t relations as interesting as objects/terms, and in fact, perhaps, their obverse? That is, I think that both objects/terms AND relations withdraw, are sites of potential surprise. Or at least, I think that the notion that objects withdraw is a very interesting thing, and that if so, the notions that relations withdraw is likely a very interesting thing as well. That is, we have new lenses with which to see the world. And then the question becomes, well, what do we see, and what does that all mean?

Whether or not new lenses allow for the formation of an entire philosophical system, a worldview, is a whole different story. The difference between lens and system, between what Deleuze might call an ‘image of thought’ versus an assemblage of ‘concepts.’ But it does raise the question for me. Why do we DO philosophy? Is it to see new things? Is it to forge conceptual apparatuses? What are they for?

However we deal with these questions, it seems to me that the very binary in question here, relations versus terms, is problematic. What do we get from dividing the world between relations and terms? Are there other binaries that are useful? When I look at the world, I see both-and, and neither-nor. But rarely do I see one or the other, for I believe that these terms, in many instances and situations, include each other.

I believe that each binary, each way of viewing the world, shows us something we hadn’t seen before. And I believe that the stuff of the world is infinite – there are an infinity of lenses whereby we can look at the world, and none are truer than others, but some fit differently than others.

That is, my view of the world is just a bit fuzzier and shifting than that implied by a world divided into relations and terms. This binary is useful in regard to particular polemics. But is it universally useful? I’m not sure I believe in ‘universally useful’. I believe rather in that which is better or worse in a game that is continually shifting around us. An experimental pragmatics, to use a term employed by Guattari. Or, to quote an old chestnut, ‘Its not if you win or lose, its how you play the game.’ I’d rework this a tiny bit to say, however, ‘Its not if its true or false, its how it plays the game.’

So, I’ll end on a Zizekian note. Relations or terms? As Zizek would say when encountering what he considers a false binary – Yes Please!